Solve for $X$. $\left[\begin{array}{rr}2 & 2 & -7 \\ 8 & 5 & 1 \\3 &12 &-15\end{array}\right]-X=\left[\begin{array}{rr}0 & 6 & 15 \\ -9 & -18 & 19 \\12 &-7 &3\end{array}\right] $ $X=$
Solution: The Strategy First, we can represent the matrices of the equation with letters, which will make the equation easier to handle. Then we can solve the equation for $X$ and obtain an expression with the letters we defined. Finally, we can substitute back the actual matrices into the resulting expression and simplify it. Solving the equation for $X$ We are given the following equation. $\left[\begin{array}{rr}2 & 2 & -7 \\ 8 & 5 & 1 \\3 &12 &-15\end{array}\right]-X=\left[\begin{array}{rr}0 & 6 & 15 \\ -9 & -18 & 19 \\12 &-7 &3\end{array}\right] $ Let's represent the above matrices as follows. $A=\left[\begin{array}{rr}2 & 2 & -7 \\ 8 & 5 & 1 \\3 &12 &-15\end{array}\right] ~~~~~~~~~ B = \left[\begin{array}{rr}0 & 6 & 15 \\ -9 & -18 & 19 \\12 &-7 &3\end{array}\right] $ Then we can rewrite the equation as follows. $A-X=B$ Now it's simple to solve the equation for $X$. $\begin{aligned}A-X&=B\\\\ A&=B+X\\\\ X&=A-B \end{aligned}$ Finding $X$ We found that $X=A-B$. Now we can substitute the actual matrices back into the expression and simplify. $\begin{aligned}X&=A-B \\\\&=\left[\begin{array}{rr}2 & 2 & -7 \\ 8 & 5 & 1 \\3 &12 &-15\end{array}\right]-\left[\begin{array}{rr}0 & 6 & 15 \\ -9 & -18 & 19 \\12 &-7 &3\end{array}\right] \\\\\\&=\left[\begin{array}{rr}(2-0) & (2-6) & (-7-15) \\ (8+9) & (5+18) & (1-19) \\(3-12) &(12+7) &(-15-3)\end{array}\right] \\\\\\&=\left[\begin{array}{rr}2 & -4 & -22 \\ 17 & 23 & -18 \\-9 &19 &-18\end{array}\right]\end{aligned}$ Summary $X=\left[\begin{array}{rr}2 & -4 & -22 \\ 17 & 23 & -18 \\-9 &19 &-18\end{array}\right]$